IEEE Signal Processing - March 2018 - 52

Although implementing fixed (orthogo(see Figure 1). These shots are usually
Wireless communication
nal)
transforms is generally fast, signal
generated by explosions or devices such
faces great challenges
adaptive transforms or dictionaries can
as air guns [1]. However, seismic signals
in light of the enormous
achieve more compact and sparse or inforhave completely different characteristics
amounts of data that
mative representation of the signal, which
from the typical voice, image, or video
must be transmitted from
makes it attractive for applications such as
signals, where lossy and lossless compression algorithms have been developed
geophones to on-site data denoising [12], [13] and compression of images and audio signals [14]-[17]. At first
and commonly used. Seismic signals gencollection centers.
glance, using an overcomplete dictionary
erally have a high dynamic range that may
may seem to degrade the compression gain
exceed 100 dB. This requires analog-toas the number of coefficients to represent a block of signals is
digital (A/D) converters with at least 24 bits. Further, the
increased. However, an overcomplete dictionary gives freedom
source shot is not stationary. Its shape and bandwidth change
in choosing the coefficients. By exploiting this property, one can
with travel time, which, in conjunction with the wavefront
aim at sparsifying the majority of coefficients and ignore them
divergence and frequency attenuation, modify the characfor signal compression.
teristics of the received signal with the travel time [2]. As a
Let X = [x 1, f, x R], x i ! R M, be a batch (collection) of
result, the seismic signals are highly nonstationary. Finally,
they exhibit decaying oscillatory patterns and are usually
data. The general goal in dictionary learning (DL) is finding
accompanied by coherent noise. These characteristics rea dictionary D and coefficients W = [w 1, f, w R], w i ! R N
quire developing dedicated compression techniques. To acsuch that W is a good informative representation of data
cess various seismic related resources such as public data
and X . DW. For example, in sparse DL, dictionary and
and seismic terminology, visit https://wiki.seg.org.
coefficients are often found using the following optimization problem:

Dictionary-based methods

In transform-based compression algorithms, a representation of the signal in a new domain is found such that it would
become more suitable for compression [3]. Early attempts for
seismic compression were mainly based on fixed transforms
such as discrete cosine transform, wavelet, and Walsh-Hadamard [4]-[6]. Wavelets and wavelet packets have been used
for both one-dimensional seismic data and stacks of seismic
data sections (e.g., [7]-[9]). In [10], locally adaptive wavelet
transform is used to gain better compression. Generalized
lapped orthogonal transforms were also optimized for seismic
data compression [11] and, when combined with embedded
zero-tree coding, they provided superior performance to traditional block-oriented algorithms or wavelets.

Server/Gateway
Node

Received Traces
from Different
Shots
Wireless Sensor

Reflection
Paths

Source Shots
(Explosions)

Gas
Oil

Figure 1. A schematic of wireless acquisition of seismic signals. For
shots at proximities of each other, but at different times, the received
signals at the sensor are correlated due to the similarities between the
propagation paths from sources to the sensor (some are depicted by
red-dashed lines). There are similar spatial correlation among traces
from the same shot recorded at sensors that are a short distance away.

t i = argmin w 0, s.t x i - Dw
w
w

D = argmin X - DW
D

2
F,

2
2

# f 6i
(1)

where f is a constant that controls the approximation error.
This optimization problem is often solved by two alternating
stages where we simply fix one variable and solve for the other.
For example, the sparse coefficients W for a fixed dictionary
D can be efficiently solved via sparse coding algorithms such
as orthogonal matching pursuit and order recursive matching
pursuit [18]. For a fixed W, the dictionary can be learned by
the method of optimal directions (MOD) [19] or K-SVD [20].
DL algorithms are generally computationally intensive and,
hence, naive implementation of ordinary optimization algorithms to solve (1) at the sensors would be costly. Since seismic signals are highly nonstationary, it is not helpful to learn a
single dictionary from few seismic traces and use it to encode
the remaining traces at the sensor.
We propose two approaches to overcome these challenges:
1) sparse-incremental online DL (SIODL) and 2) rate-optimized DL. While the former approach is significantly faster
than the traditional DL, it performs reasonably well and is
suitable for deploying in sensors. The latter, although it has
better performance, is computationally more complex. Hence,
we suggest using it at the gateway node or server that generally has more computational power. Moreover, upon receiving more data from sensors, the gateway node can update the
dictionary and broadcast it to all sensors at a negligible communication cost.

SIODL
Conventional DL algorithms rely on alternate optimization between the two subproblems in (1). Hence, they are

IEEE Signal Processing Magazine

March 2018

https://wiki.seg.org

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